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By Osamu Watanabe

The mathematical thought of computation has given upward thrust to 2 very important ap­ proaches to the casual thought of "complexity": Kolmogorov complexity, usu­ best friend a complexity degree for a unmarried item akin to a string, a chain etc., measures the quantity of data essential to describe the item. Compu­ tational complexity, often a complexity degree for a collection of gadgets, measures the compuational assets essential to realize or produce components of the set. The relation among those complexity measures has been thought of for greater than twenty years, and will fascinating and deep observations were got. In March 1990, the Symposium on conception and alertness of minimum­ size Encoding was once held at Stanford college as part of the AAAI 1990 Spring Symposium sequence. a few periods of the symposium have been devoted to Kolmogorov complexity and its family members to the computational complexity the­ ory, and perfect expository talks got there. Feeling that, because of the significance of the fabric, a way will be discovered to percentage those talks with researchers within the desktop technology group, I requested the audio system of these periods to put in writing survey papers in response to their talks within the symposium. In reaction, 5 audio system from the classes contributed the papers which look during this book.

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Extra resources for Kolmogorov Complexity and Computational Complexity

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Continuous optimization problems and a polynomial hierarchy of real functions. J. Complexity 1:210-231, 1985. K. Ko. On the notion of infinite pseudorandom sequences: Theoret. Comput. Sci. 39:9-33, 1986. K. Ko and U. Schoning. On circuit-size complexity and the low hierarchy in N P. SIAM J. Computing 14:41-51, 1985. S. Kurtz. On the random oracle hypothesis. Info. and Control 57:40-47, 1983. R. Ladner, N. Lynch, and A. Selman. A comparison of polynomial-time reducibilities. Theoret. Comput. Sci.

In §6, we apply these results to our main topic, which is the complexity and distribution of ~;,-hard problems for ESPACE. It is well-known that such problems are not feasibly decidable and must obey certain lower bounds on their complexities. , > 2n ' for some E > 0) space-bounded Kolmogorov complexity; and Orponen and Schoning[OS86] have (essentially) proven that every ~;,-hard language for ESPACE has a dense DSPACE(2 cn )complexity core. Intuitively, such results are not surprising, as we do not expect hard problems to be simple.

1st Conference on Structure in Complexity Theory, SpringerVerlag, Lecture Notes in Computer Science 223:23-34, 1986. T. Baker, J. Gill, and R. Solovay. Relativizations ofthe P =? NP question. [BGS75] SIAM J. Computing 4:431-442,1975. [BB86] J. zar and R. Book. Sets with small generalized Kolmogorov complexity. Acta Informatica 23:679-688, 1986. [BBS86a] J. Balcazar, R. Book, and U. Schoning. The polynomial-time hierarchy and sparse oracles. J. Assoc. Comput. Mach. 33:603-617,1986. [BBS86b] J. Balcazar, R.

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